RULE VI. § 45. To Multiply by any Number exceeding 12, and containing two or more SIGNIFICANT FIGURES. 1. Multiply by each significant figure, separately, of the multiplier,—placing the several rows of products one under another, with the first figure of each product under the multiplying figure,—and, in that order, add the several products together for the entire product. 2. Ciphers in the right of either or both of the factors, are omitted in multiplying; but as many Os must be placed in the right of the product. EXAMPLES. 1. To Multiply 8072 by 39; that is, to find 39 times 8072. 8072 72648 24216 314808 Multiplying first by 9, we say 9 times 2 is 18, and set the first product figure 8 under the multiplying figure 9; then, 9 times 7 is 63, and 1 makes 64; 9 times 0 is 0, and 6 makes 6, &c. Multiplying next by 3, we say 3 times 2 is 6, and set the 6 under the multiplying figure 3; 3 times 7 is 21; 3 times 0 is 0, and 2 is 2, &c. Adding the two rows of product figures together, in the order in which they stand, we say 8 is 8; 6 and 4 are 10, &c. The entire Product is 314808. 2. To Multiply 8420 by 30900; or to find 30900 times 8420; 8420 30900 7578 2526 260178 000 Omitting Os in the right of both factors, we multiply 842 by 309, setting the first figures, 8 and 6, of the products under the multiplying figures, 9 and 3, respectively. To the product thus found, we annex three Os, for the Os omitted in multiplying. When the multiplying figure is tens; or hundreds, &c., the first product figure is set under tens, or hundreds, &c., respectively, to increase the product in the same degree in which the multiplying figure is increased in value, by distance from the unit's place (§ 16). In the first example, 2 multiplied by 3 produces 6. But the 3 is 3 tens, or 10 times the simple 3; the 6 must therefore be 10 times the simple 6; and this tenfold value is assigned to 6 by setting it under the 3 tens, and adding it in the ten's column. In the second example, we multiplied 842 by 309. In 309 the 3 is 3 hundreds, or 100 times the simple 3. The first product figure 6 must therefore be 100 times the simple 6; and this hundredfold value is assigned to 6 by setting it under the 3 hundreds, and adding it in the hundreds column. The first product figure being set in its proper place, the second, third, &c., fall in their proper places, in ascending orders of units towards the left.—The sum of the partial products is the entire product (§ 23). As already shown, in connexion with Rule V, for each 0 omitted in the right of either factor, a 0) must be placed in the right of the product. EXERCISES. 41. Find the Product 24X360730. 42. Find the Product 307×80379. 43. Find the Product 5372×7684. 44. Find the Product 80760X870. 45. Find the Product 13X730000. 46. Find the Product 740X87305. 47. Find the Product 9034×8076. 48. Find the Product 30407307. 49. Find the Product 400070×990. Ans. 8657520 Ans. 396069300. 50. The number of yards in a mile being 1760, how many yards are there in 15 miles? The number of yards in 15 miles is 15 times 1760 yards. 51. There are 24 hours in one day. are there in a year of 365 days? Ans. 26400 yards. How many hours then The number of hours in a year is 365 times 24 hours; but 24 times 365 will produce the same number, and it is more convenient to make the less number the multiplier. Ans. 8760 hours. 52. A hogshead of wine or brandy contains 63 gallons. How many gallons would there be in 250 hogsheads? Ans. 15750 gallons. Ans. 32895 dollars. 53. What sum should be paid for a plantation containing 765 acres, at 43 dollars per acre? 54. If a man can walk 35 miles in a day, how far could he walk, at that rate, in a year, or 365 days? Ans. 12775 miles. 55. A has 340 acres of land worth 18 dollars an acre; and B has 239 acres worth 22 dollars an acre. How many acres have the two together, and what is the value of the whole. Ans. 579 acres; and 11378 dollars. 56. A merchant bought 475 barrels of flour, at 15 dollars a barrel. He sold 280 barrels of it, at 16 dollars, and the rest at 14 dollars a barrel; what did he gain or lose? Ans. gained 85 dollars. 57. One manufacturer exported 234 bales of cotton cloth,each bale containing 2400 yards; another exported 370 bales, each containing 1050 yards. Which of them exported the greater quantity, and by how many yards? Ans. The first, 173100 yards. 58. Farmer A had in wheat 205 acres, which produced 27 bushels per acre. Farmer B had 320 acres, which produced 19 bushels per acre. What quantity of wheat was raised by them both. Ans. 11615 bushels. 59. A speculator bought 150 head of cattle, and 47 mules. He made a profit of 13 dollars a head on the former, and 17 on the latter; what was gained by the speculation? Ans. 2749 dollars. 60. Bought 360 acres of land, at 35 dollars per acre; and at another time double that quantity, at double the price per acre. What was the whole quantity of land purchased, and the sum paid for it? Ans. 1080 acres; and 63000 dollars. 61. Two persons start together from the same place, and travel in the same direction. One proceeds at the rate of 29 miles per day, and the other at the rate of 31 miles per day. What distance will be between them at the end of 25 days? Ans. 50 miles. 62. A merchant bought 18 bales of linen, each containing 22 pieces, and each piece containing 40 yards. How many pieces and how many yards did he buy? Ans. 396 pieces; and 15840 yards. 63. In a certain orchard there are 30 rows of apple trees, with 44 trees in each row. Allowing 2500 apples to each tree what number of apples would there be in the orchard? Ans. 3300000 apples. 64. A farmer bought three tracts of land. The first and second contained each 280 acres, and the third as many as both the other two; how many acres did the farmer purchase, and what did the whole amount to, at 33 dollars per acre? Ans. 1120 acres; and 36960 dollars. 65. Allowing a person's annual income to be 5000 dollars, and his daily expenses 3 dollars, what would be the amount of his annual saving,—there being 365 days in a year? Ans. 3905 dollars. 66. If 327 head of cattle were purchased at 13 dollars & head, and 405 were purchased at 11 dollars a head, what would be the profit or loss on the whole at 12 dollars a head? Ans. Profit, 78 dollars. 67. A planter sold 139 bales of cotton, at an average of 32 dollars per bale, and out of the proceeds bought 29 mules, at 49 dollars each, and 4 pair of oxen, at 52 dollars a pair; what sum had he left from the sale of his cotton ? Ans. 2819 dollars. 68. A sends 209 tons of coal to New York city; B sends as much as A, wanting 10 tons; and C sends as much as A and B together. What was each man's proceeds of sale, at 13 dollars per ton? Ans. A's 2717 dollars; B's 2587; C's 5304. 69. The President of the United States receives a salary of 25 thousand dollars a year. To what sum does his salary amount in 4 years, or one presidential term? Ans. 100000 dollars. 70. The circumference of the Earth is about 25 thousand miles, and the distance to the Sun is 3 thousand 8 hundred times the Earth's circumference. What then is the distance to the Sun ? Ans. 95000000 miles. 71. The velocity of light is 192 thousand 500 miles per second. Through what distance then does light move in one minute, which is 60 seconds! Ans. 11550000 miles. 72. The Earth turns around its axis once in every 24 hours, and moves 68 thousand miles an hour in its orbit around ti e Sun. How far then are we carried along the Earth's or. .t during one revolution of the Earth on its axis? Ans. 1632000 mile: DIVISION. $46. DIVISION consists in finding how many times a greater number contains a less, or what part a less number is of a greater. The number to be divided is called the dividend; the dividing number the divisor; and the number or part found, the quotient. One half is one of the two equal parts,-two thirds are two of the three equal parts,—and so on, into which any quantity may be divided. What is meant by one third? By one fifth? By two fifths? By one fourth? By three fourths? When we say 2 is contained in 6, 3 times, we divide 6 by 2. 6 is the dividend, 2 the divisor, and 3 the quotient. Also, 2 is one third of 6, because if 6 were divided into three equal parts, each part would be 2. 3 is what part of 6? 4 is what part of 12? 5 is what part of 20? How many times is 3 contained in 6 How many times is 4 contained in 12? How many times is 5 contained in 20? If the dividend be 24, and the divisor 4, what will the quotient be? If the dividend be 35, and the divisor 5? If the dividend be 42, and the divisor 7? If the dividend be 56, and the divisor 8? $47. The Quotient of a less number divided by a greater, is the part that the less is of the greater; and is denoted by the less over the greater, with a line between them. Thus 1 divided by 2 is one half, because 1 is one half of 2. How much is 1 divided by 3; that is, 1 is what part of 3? 2 divided by 7? 3 divided by 5? Subtraction and Division. $48. The subtraction of a less number from a greater, repeatedly, is equivalent to dividing the greater by the less, because it shows how many times the greater contains the less. Thus 5 from 15 leaves 10, 5 from 10 leaves 5, and 5 from 5 leaves 0; so that 5 may be subtracted 3 times from 15, or is contained 3 times in 15. |